An adaptive finite element method in reconstruction of coefficients in Maxwell's equations from limited observations

We propose an adaptive finite element method for the solution of a coefficient inverse problem of simultaneous reconstruction of the dielectric permittivity and magnetic permeability functions in the Maxwell's system using limited boundary observations of the electric field in 3D. We derive a posteriori error estimates in the Tikhonov functional to be minimized and in the regularized solution of this functional, as well as formulate corresponding adaptive algorithm. Our numerical experiments justify the efficiency of our a posteriori estimates and show significant improvement of the reconstructions obtained on locally adaptively refined meshes.Comment: Corrected typo

Fast Microwave Tomography Algorithm for Breast Cancer Imaging

Microwave tomography has shown promise for breast cancer imaging. The microwaves are harmless to body tissues, which makes microwave tomography a safe adjuvant screening to mammography. Although many clinical studies have shown the effectiveness of regular screening for the detection of breast cancer, the anatomy of the breast and its critical tissues challenge the identification and diagnosis of tumors in this region. Detection of tumors in the breast is more challenging in heterogeneously dense and extremely dense breasts, and microwave tomography has the potential to be effective in such cases. The sensitivity of microwaves to various breast tissues and the comfort and safety of the screening method have made microwave tomography an attractive imaging technique. Despite the need for an alternative screening technique, microwave tomography has not yet been introduced as a screening modality in regular health care, and is still subject to research. The main obstacles are imperfect hardware systems and inefficient imaging algorithms. The immense computational costs for the image reconstruction algorithm present a crucial challenge. 2D imaging algorithms are proposed to reduce the amount of hardware resources required and the imaging time. Although 2D microwave tomography algorithms are computationally less expensive, few imaging groups have been successful in integrating the acquired 3D data into the 2D tomography algorithms for clinical applications. The microwave tomography algorithms include two main computation problems: the forward problem and the inverse problem. The first part of this thesis focuses on a new fast forward solver, the 2D discrete dipole approximation (DDA), which is formulated and modeled. The effect of frequency, sampling number, target size, and contrast on the accuracy of the solver are studied. Additionally, the 2D DDA time efficiency and computation time as a single forward solver are investigated.\ua0 The second part of this thesis focuses on the inverse problem. This portion of the algorithm is based on a log-magnitude and phase transformation optimization problem and is formulated as the Gauss-Newton iterative algorithm. The synthetic data from a finite-element-based solver (COMSOL Multiphysics) and the experimental data acquired from the breast imaging system at Chalmers University of Technology are used to evaluate the DDA-based image reconstruction algorithm. The investigations of modeling and computational complexity show that the 2D DDA is a fast and accurate forward solver that can be embedded in tomography algorithms to produce images in seconds. The successful development and implementation in this thesis of 2D tomographic breast imaging with acceptable accuracy and high computational cost efficiency has provided significant savings in time and in-use memory and is a dramatic improvement over previous implementations

A Discrete Dipole Approximation Forward Solver for Microwave Breast Imaging

Breast cancer has the highest incidence rate of cancers in women worldwide. Early detection results in a higher survival rate. Drawbacks in conventional imaging modalities, including painful exams, have limited periodic breast screening. Microwave to mography has the potential to be a compelling alternative or complement to other imaging techniques. Advantages of microwave tomography is that it is harmless, comfortable, and cost effective. Microwave tomography has not yet fully been translated into the clinic, even if clinical trials are ongoing.\ua0One important challenge is high computational demands of microwave tomography algorithms. 3D tomography algorithms require multiple hours and a large amount of hardware resources to produce images. 3D imaging algorithms are usually implemented and tested for simulations setup and barely used in clinical settings. 2D microwave tomography algorithms are computationally less expensive compared to 3D algorithms. Few imaging groups have been successful in integrating the acquired 3D data into the 2D tomography algorithms for clinical applications.\ua0The microwave tomography algorithms include two main computation problems; forward and inverse. The forward problem has to be solved multiple times and the resulting computational cost is the time limiting step in microwave tomography algorithms, and this thesis is devoted to addressing it.\ua0In this thesis, the two-dimensional forward problem is modelled and formulated. In particular, the two-dimensional discrete dipole approximation (DDA) is proposed as a new forward solver for microwave tomography. The accuracy of the 2D DDA with respect to sampling number, size, and contrast of target are investigated. Moreover, the 2D DDA time efficiency and computation time are studied.The forward solver computation times for direct, iterative, and iterative combined with fast Fourier transformation (FFT) solvers are calculated. The observations imply that the 2D DDA is an accurate, reliable, and rapid forward solver, and using the Krylov subspace methods combined with the FFT accelerate the computation time significantly

A discrete dipole approximation solver based on the COCG-FFT algorithm and its application to microwave breast imaging

We introduce the discrete dipole approximation (DDA) for efficiently calculating the two-dimensional electric field distribution for our microwave tomographic breast imaging system. For iterative inverse problems such as microwave tomography, the forward field computation is the time limiting step. In this paper, the two-dimensional algorithm is derived and formulated such that the iterative conjugate orthogonal conjugate gradient (COCG) method can be used for efficiently solving the forward problem. We have also optimized the matrix-vector multiplication step by formulating the problem such that the nondiagonal portion of the matrix used to compute the dipole moments is block-Toeplitz. The computation costs for multiplying the block matrices times a vector can be dramatically accelerated by expanding each Toeplitz matrix to a circulant matrix for which the convolution theorem is applied for fast computation utilizing the fast Fourier transform (FFT). The results demonstrate that this formulation is accurate and efficient. In this work, the computation times for the direct solvers, the iterative solver (COCG), and the iterative solver using the fast Fourier transform (COCG-FFT) are compared with the best performance achieved using the iterative solver (COCG-FFT) in C++. Utilizing this formulation provides a computationally efficient building block for developing a low cost and fast breast imaging system to serve under-resourced populations

Constrained FoV Radiated Power as a Figure of Merit of Phased Arrays

In this paper, we propose quantifying the radiated power of phased arrays or, in general, directive antennas, by the Constrained-View Radiated Power (CVRP). The constrained view shall be interpreted here as the Field-of-View (FoV) of an antenna that defines a region in space where focusing the radiated power is highly desired. In the limiting cases, we have that CVRP equals the Total Radiated Power (TRP) when the FoV covers the whole sphere, while, if the FoV reduces to a single point in space, the CVRP equals the Equivalent Isotropic Radiated Power (EIRP). We further present an analysis based on measured radiation patterns of a 16-element, linearly polarized, millimeter-Wave (mmWave), planar phased array antenna operating at 28 GHz. We compare the results to two ideal planar array antennas with the same number of Huygens and cosine elements. The evaluated figure of merit is computed for different scanning angles, as well as for different malfunctions of antenna elements, both for the real and simulated arrays. The results show that the introduced figure of merit could be potentially used for the detection of malfunctioning elements in antenna arrays as well as to characterize the impact of scan loss. Furthermore, CVRP is useful to straightforwardly and significantly characterize the performance of a directive antenna in terms of the power radiated towards a specific region in space

Expansion of the nodal-adjoint method for simple and efficient computation of the 2d tomographic imaging jacobian matrix

This paper focuses on the construction of the Jacobian matrix required in tomographic reconstruction algorithms. In microwave tomography, computing the forward solutions during the iterative reconstruction process impacts the accuracy and computational efficiency. Towards this end, we have applied the discrete dipole approximation for the forward solutions with significant time savings. However, while we have discovered that the imaging problem configuration can dramatically impact the computation time required for the forward solver, it can be equally beneficial in constructing the Jacobian matrix calculated in iterative image reconstruction algorithms. Key to this implementation, we propose to use the same simulation grid for both the forward and imaging domain discretizations for the discrete dipole approximation solutions and report in detail the theoretical aspects for this localization. In this way, the computational cost of the nodal adjoint method decreases by several orders of magnitude. Our investigations show that this expansion is a significant enhancement compared to previous implementations and results in a rapid calculation of the Jacobian matrix with a high level of accuracy. The discrete dipole approximation and the newly efficient Jacobian matrices are effectively implemented to produce quantitative images of the simplified breast phantom from the microwave imaging system

Vpliv razvojnega financiranja na poslovanje izbranega podjetja v času globalne finančne krize

summary:We propose an adaptive finite element method for the solution of a coefficient inverse problem of simultaneous reconstruction of the dielectric permittivity and magnetic permeability functions in the Maxwell's system using limited boundary observations of the electric field in 3D. We derive a posteriori error estimates in the Tikhonov functional to be minimized and in the regularized solution of this functional, as well as formulate the corresponding adaptive algorithm. Our numerical experiments justify the efficiency of our a posteriori estimates and show significant improvement of the reconstructions obtained on locally adaptively refined meshes

Application of the discrete dipole approximation in microwave breast imaging

In this work, we propose the two-dimensional discrete dipole approximation (2D DDA) to calculate the electric field distributions in the microwave imaging system. The motive is to develop a significantly fast reconstruction algorithm. To accomplish this, the 2D DDA on a uniform grid in the forward model zone is used, which enables computational times be significantly reduced compared to common algorithms

Integrating the discrete dipole approximation forward solver with a microwave tomography algorithm

The discrete dipole approximation (DDA) has been suggested as a viable alternative method for computing the forward solutions during the iterative reconstruction process used in microwave tomography. While efficient algorithms such as finite difference time domain and finite elements have been successfully used by multiple groups, the forward solution time remains the time-limiting step. The DDA has been shown to be accurate and efficient as a forward solver. However, the configuration of the imaging scenario can have a significant impact on its efficiency. We examine two possible forward solution set-ups and describe the benefits with respect to implementing the DDA